Ted. From the way you are describing the issue, it sounds like Marine electrical systems are a new thing. If so you should either seek an electrical marine professional, take a class or you have to do a bit of reading to understand the electrical systems on your boat.
The Battery is part of the DC system. It can store power from either the Alternator on the engine or the Charles 30 amp charger you identify.
The other source of power is from the plug you run ashore to the dock power plug. This dock connection ( in the big yellow or orange extension cord) provides 120 Volt 30 amp AC electrical power. In the boat it controled by the AC panel.
inverter always reads 0, fridge at 60 degrees, batteries 1.3 years old.
An inverter takes DC 12 V power and inverts it into AC 120 V power.
The Charger (which could be in the same unit called an Charge/Inverter - have no idea what you have) Takes AC 120V power from the shore and changes it to 12V DC power to charge your batteries.
There will be AC or DC breakers on the appropriate panel to turn these units on or off. (A picture of your panels at the Nav station would be helpful.)
This does not answer the specific question you posted, I know. We need to know from you which of the units are not working. Pictures of your system and machine labels are a good place to start.
Here is one book to start you understanding.
Boatowner′s Mechanical and Electrical Manual, third edition
Nigel Calder
Maintain, repair, and improve a boats essential systems. A comprehensive guide to electrical, mechanical, and propulsion systems.